M₁₃ Game

Here's a playable version of a puzzle invented by John Conway, in the spirit of the Rubik's cube or the "15 puzzle" made famous by Sam Loyd. There are some links at the bottom of the page to an explanation of where the puzzle comes from and how it's related to a mathematical object called the "Mathieu groupoid", M₁₃ — but you don't need to know any of that to play the game!

How to Play

We have a board with 13 positions and 12 tokens, with one position left empty (the "hole", marked as a red circle). The positions are connected together in blocks of four, as described below. A move of the game consists in:

• picking a token to move
• finding the block it shares with the hole
• jumping the token to the hole
• swapping the tokens in the other two positions in the block

Clicking at random a few times quickly shuffles the tokens around. Just like a Rubik's cube, the challenge is to unshuffle them back to their original positions! It's easy to find some very simple patterns:

• moving the same token again undoes the previous move
• more generally, moving repeatedly within the same block only shuffles the tokens in that block
• moving the same pair of tokens three times (e.g. 1,2,1,2,1,2) is the identity operation

Projective board

The board for this game is based a diagram drawn by Bob Harris and linked in the n-Category Cafe article below.

The 13 positions on the board are grouped into blocks of 4, indicated by the lines: some straight lines like the one through positions 11, 4, 5, and 6; and some sweeping arcs like the one through positions 7, 6, 2, and 10. (There are 13 lines in total — can you find them all?)

Just like in ordinary Euclidean geometry, any two points lie on exactly one common line and any two lines meet at exactly one point. But in this geometry every point lies on exactly 4 lines and each line contains exactly 4 points!

Links

I first learned about this from an article at the n-Category Cafe by John Baez, about a groupoid that John Conway called the "Mathieu groupoid", M₁₃.

Conway, Elkies and Martin describe a way of understanding this groupoid in terms of a "sliding block puzzle" game played on the projective plane over F₃. As John Baez explains, the groupoid M₁₃ "consists of the permutations of counters that can be obtained by composing finite sequence of moves". The sequences of moves that leave the hole in its original position form a subgroupoid that turns out to be the Mathieu group, M₁₂.

In 2007 Lieven Le Bruyn wrote up an explanation of the sliding block puzzle in a way that transposed the action to a circular board. He mentions there a playable version of the game made by Sebastian Egner, which unfortunately no longer seems to be available.